• 论文 •

### 高波数Helmholtz方程的有限元方法和连续内罚有限元方法

1. 南京大学数学系, 南京 210093
• 收稿日期:2017-08-31 出版日期:2018-06-15 发布日期:2018-05-15
• 基金资助:

国家自然科学基金（11525103，91630309，11621101）资助.

Wu Haijun. FEM AND CIP-FEM FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBER[J]. Mathematica Numerica Sinica, 2018, 40(2): 191-213.

### FEM AND CIP-FEM FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBER

Wu Haijun

1. Department of Mathematics, Nanjing University, Nanjing 210093, China
• Received:2017-08-31 Online:2018-06-15 Published:2018-05-15

Finite element methods (FEM) and continuous interior penalty finite element methods (CIP-FEM) are considered for the Helmholtz equation with high wave number. Preasymptotic stability and error analyses with explicit dependence on the wave number k are provided for the linear versions of the methods. Three approaches for the analyses will be introduced. In order to reduce greatly the pollution error of the methods, the choice of the penalty parameters for the CIP-FEM will be discussed. Numerical examples are given to verify the theoretical results.

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 [1] Ainsworth M. Discrete dispersion relation for hp-version finite element approximation at high wave number[J]. SIAM J. Numer. Anal., 2004, 42(2):553-575.[2] Aziz A and Kellogg R. A scattering problem for the Helmholtz equation. In Advances in Computer Methods for Partial Differential Equations-Ⅲ, 1979, 93-95.[3] Babuška I and Sauter S. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?[J]. SIAM Rev., 2000, 42(3):451-484.[4] Brenner S and Scott R. The Mathematical Theory of Finite Element Methods, volume 15. Springer Science & Business Media, 2007.[5] Burman E, Zhu L and Wu H. Linear continuous interior penalty finite element method for Helmholtz equation with high wave number:One-dimensional analysis[J]. Numer. Meth. Par. Diff. Equ., 2016, 32:1378-1410.[6] Chen H, Lu P and Xu X. A hybridizable discontinuous Galerkin method for the Helmholtz equation with high wave number[J]. SIAM J. Numer. Anal., 2013, 51:2166-2188.[7] Chen H, Wu H and Xu X. Multilevel preconditioner with stable coarse grid corrections for the helmholtz equation[J]. SIAM J. Sci. Comput., 2015, 37:A221-A244.[8] Chen Z, Wu T and Yang H. An optimal 25-point finite difference scheme for the Helmholtz equation with PML[J]. J. Comput. Appl. Math., 2011, 236:1240-1258.[9] Chen Z and Xiang X. A source transfer domain decomposition method for Helmholtz equations in unbounded domain[J]. SIAM J. Numer. Anal., 2013, 51(4):2331-2356.[10] Demkowicz L, Gopalakrishnan J, Muga I and Zitelli J. Wavenumber explicit analysis of a DPG method for the multidimensional Helmholtz equation[J]. Comput. Methods Appl. Mech. Engrg., 2012, 214(12):126-138.[11] Deraemaeker A, Babuška I and Bouillard P. Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions[J]. Internat. J. Numer. Methods Engrg., 1999, 46:471-499.[12] Douglas J and Dupont T. Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods. Lecture Notes in Physics, 1976, 58:207-216.[13] Du Y and Wu H. An improved pure source transfer domain decomposition method for Helmholtz equations in unbounded domain. ArXiv e-prints, 2015.[14] Du Y and Wu H. Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number[J]. SIAM J. Numer. Anal., 2015, 53(2):782-804.[15] Du Y and Zhu L. Preasymptotic error analysis of high order interior penalty discontinuous Galerkin methods for the Helmholtz equation with high wave number[J]. J. Sci. Comput., 2016, 67:130-152.[16] Engquist B and Runborg O. Computational high frequency wave propagation[J]. Acta numerica, 2003, 12:181-266.[17] Engquist B and Ying L. Sweeping preconditioner for the Helmholtz equation:hierarchical matrix representation[J]. Comm. Pure Appl. Math., 2011, 64(5):697-735.[18] Engquist B and Ying L. Sweeping preconditioner for the Helmholtz equation:moving perfectly matched layers[J]. Multiscale Model. Simul., 2011, 9:686-710.[19] Feng X and Wu H. Discontinuous Galerkin methods for the Helmholtz equation with large wave numbers[J]. SIAM J. Numer. Anal., 2009, 47(4):2872-2896.[20] Feng X and Wu H. hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number[J]. Math. Comp., 2011, 80(276):1997-2024.[21] Feng X and Wu H. An absolutely stable discontinuous Galerkin method for the indefinite timeharmonic Maxwell equations with large wave number[J]. SIAM J. Numer. Anal., 2014, 52:2356-2380.[22] Hiptmair R, Moiola A and Perugia I. Plane wave discontinuous Galerkin methods for the 2d Helmholtz equation:Analysis of the p-version[J]. SIAM J. Numer. Anal., 2011, 49:264-284.[23] Hiptmair R, Moiola A and Perugia I. A Survey of Trefftz Methods for the Helmholtz Equation, pages 237-279. Springer International Publishing, Cham, 2016.[24] Hu Q and Zhang H. Substructuring preconditioners for the systems arising from plane wave discretization of Helmholtz equations[J]. SIAM J. Sci. Comput., 2016, 38:A2232-A2261.[25] Ihlenburg F. Finite element analysis of acoustic scattering, volume 132 of Applied Mathematical Sciences. Springer-Verlag, New York, 1998.[26] Ihlenburg F and Babuska I. Finite element solution of the Helmholtz equation with high wave number part Ⅱ:the hp version of the FEM[J]. SIAM J. Numer. Anal., 1997, 34(1):315-358.[27] Ihlenburg F and Babuška I. Finite element solution of the Helmholtz equation with high wave number. I. The h-version of the FEM[J]. Comput. Math. Appl., 1995, 30(9):9-37.[28] Douglas J, Santos J and Sheen D. Approximation of scalar waves in the space-frequency domain[J]. Math. Models Methods Appl. Sci., 1994, 4:509-531.[29] Li Y and Wu H. Fem and cip-fem for helmholtz equation with high wave number and pml truncation. Submitted, 2017.[30] Melenk J, Parsania A and Sauter S. General DG-methods for highly indefinite Helmholtz problems[J]. J. Sci. Comp., 2013, 57(3):536-581.[31] Melenk J and Sauter S. Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions[J]. Math. Comp., 2010, 79(272):1871-1914.[32] Melenk J and Sauter S. Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation[J]. SIAM J. Numer. Anal., 2011, 49(3):1210-1243.[33] Moiola A and Spence E. Is the Helmholtz equation really sign-indefinite?[J]. SIAM Rev., 2014, 56:274-312.[34] Schatz A. An observation concerning Ritz-Galerkin methods with indefinite bilinear forms[J]. Math. Comp., 1974, 28:959-962.[35] Shen J and Wang L. Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains[J]. SIAM J. Numer. Anal., 2007, 45(5):1954-1978.[36] Thompson L. A review of finite-element methods for time-harmonic acoustics[J]. J. Acoust. Soc. Amer., 2006, 119:1315-1330.[37] Wang K and Wong Y. Is pollution effect of finite difference schemes avoidable for multi-dimensional Helmholtz equations with high wave numbers?[J]. Commun. Comput. Phys., 2017, 21:490-514.[38] Wu H. 高波数helmholtz方程的内罚有限元方法[J].中国科学:数学, 2012, 42:429-444.[39] Wu H. Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part I:linear version[J]. IMA J. Numer. Anal., 2013, 34(3):1266-1288.[40] Zhu L and Wu H. Preasymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. part Ⅱ:hp version[J]. SIAM J. Numer. Anal., 2013, 51(3):1828-1852.[41] Zienkiewicz O. Achievements and some unsolved problems of the finite element method[J]. Int. J. Numer. Meth. Engng., 2000, 47:9-28.
 [1] 卢培培, 许学军. 高波数波动问题的多水平方法[J]. 计算数学, 2018, 40(2): 119-134. [2] 王坤, 张扬, 郭瑞. Helmholtz方程有限差分方法概述[J]. 计算数学, 2018, 40(2): 171-190. [3] 骆其伦, 黎稳. 二维Helmholtz方程的联合紧致差分离散方程组的预处理方法[J]. 计算数学, 2017, 39(4): 407-420. [4] 郑权, 高玥, 秦凤. Helmholtz方程外边值问题的基于修正的DtN边界条件的有限元方法[J]. 计算数学, 2016, 38(2): 200-211. [5] 孟文辉, 王连堂. Helmholtz方程周期Green函数及其偏导数截断误差收敛阶的分析[J]. 计算数学, 2015, 37(2): 123-136. [6] 张敏,杜其奎,. 椭圆外区域上Helmholtz问题的自然边界元法[J]. 计算数学, 2008, 30(1): 75-88. [7] 杨超,孙家昶. 一类六边形网格上拉普拉斯4点差分格式及其预条件子[J]. 计算数学, 2005, 27(4): 437-448. [8] 贾祖朋,邬吉明,余德浩. 三维Helmholtz方程外问题的自然边界元与有限元耦合法[J]. 计算数学, 2001, 23(3): 357-368. [9] 余德浩,贾祖朋. 二维Helmholtz方程外问题基于自然边界归化的非重叠型区域分解算法[J]. 计算数学, 2000, 22(2): 227-240.